Short and Long Distance Effects in the Decay $τ\to πν_τ(γ)$

Abstract

We calculate the radiative corrections to the decays $τ\to M ν_τ$ and $π\to l ν_ł$, where the meson $M$ is $M=π$ or $K$ and the lepton $l$ is $l = e$ or $μ$. We perform a complete calculation, which includes internal bremsstrahlung and structure dependent radiation in the radiative decays and point meson, hadronic structure dependent and short distance contributions in the virtual corrections. Our result for the radiative correction to the ratio $Γ(τ\toπν_τ(γ))/ Γ(π\toμν_μ(γ))$ is $δR_{τ/π} = \left(0.16_{\T - 0.14}^{\T+0.09}\right) \%$. For the ratio $Γ(τ\to Kν_τ(γ)) / Γ(K\to μν_μ(γ))$, we obtain $δR_{τ/K } = \left(0.90_{\T - 0.26}^{\T + 0.17}\right) \%$. For completeness we have also calculated the ratio of the electronic and muonic decay modes of the pion.